How do I paint this on a coffee mug?

you can buy a fixative that you spray on after painting and it helps reduce damage by sun/hot water etc. I dont think your friends will be able to machine-wash them, but thats okay.

for designs, you could paint what they like or what you associate them with, like cats, dogs, the beach, flowers etc. and then of course their names. try and colour code them, like if its a bech theme, then use only blue and yellow/sandy colours so it dosent look messy.

you can buy shape stamp things that you dip in paint and press on the cups.

if you can, try and get cups that havent yet been fired in a kiln, so like, not glossy. but if you can’t find them thats all good. Acrylic paints will work good. dont add too much water as the paint will run. and practice first on paper what designs your doing, and maybe buy 2-3 extra cups, one for experimenting with the feel/surface/texture etc and one or two in case of mistakes/dropping and smashing. if you do do a mistake, if you are quick, you can just wipe the whole thing off and wash the cup if it is shiny, but if it is matte then its stuck there- you could paint over it though.

have fun :)